ABOUT THE AUTHOR Torben Larsen received his M.Sc. degree in electrical engineering from Aalborg University, Denmark, in 1988. In 1995 he founded the RF Integrated Systems and Circuits group, which today counts more than 10 researchers and technicians. He received the Dr.Techn. degree from Aalborg University in 1998. In 1999 he received the Spar Nord research Prize. Since 2001 he has been working as professor at Aalborg University. He has been project leader for several research projects, and has close co-operation with several mobile phone companies. Areas of specialized research interests include circuit theory in general, noise theory, nonlinear analysis techniques, RF CMOS technology, and digital modulation techniques.

Spectral regrowth is a significant distortion mechanism in nonlinear devices, particularly for power amplifiers handling varying envelope signals in applications such as wireless. One way of analyzing this distortion is to use approximate discrete spectrum representations of the signals. However, compared to the generally correct continuous-spectrum description, the discrete spectrum approach introduces errors in both output bandwidth estimation and in power estimation of the spectral regrowth. This paper shows that the bandwidth error with the discrete spectrum method can be rather large but easily predictable. The power error is relatively small for practical cases"an example of a CDMA signal introduces an error of less then 1% when using 11 frequency points to represent a 1.25 MHz signal.

CDMA signals are increasing in wireless communication to facilitate high data rates. CDMA is presently used in the IS-95 system, and it is included in the European IMT-2000 proposal. Spectral regrowth, a key issue for these systems, is a phenomena that occurs if a band-limited time-varying envelope signal is passed through an odd-order nonlinear circuit. In this situation, the output spectrum is wider than the input spectrum"spectral regrowth"and co-channel and adjacent-channel distortion results. Discrete-spectrum approaches are popular techniques for analyzing spectral regrowth phenomena. However, these approaches may produce errors compared to continuous-spectrum signals with respect to spectral bandwidth and spectral regrowth power level. This paper investigates some of the implications of this error introduction.

In general, the signals involved in spectral-regrowth analysis are, more or less, random and continuous, and thus have a continuous frequency-domain representation. This is not the case when you use a Fourier analysis that produces approximate techniques. This type of signal has a discrete spectrum description. Reference uses a Fourier-series approach to represent the CDMA signal in IS95. Using a Fourier series means using a number of discrete frequency points to approximate the signal, which is generally continuous in a certain bandwidth 2B.

Considering a deterministic and discrete spectrum approach, such as the one in Maas, gives the result in Figure 1a. In this figure, the input is consists of the sum of three cosine signals, which results in a discrete input-frequency spectrum with components at the input frequencies. The distortion components are also illustrated in Figure 1a for a third-order nonlinear device. Observe that the input consists of the three tones, but there are only two tones in the upper and lower spectral regrowth bands at and below f₀ - 2Df, and at and above f₀ + 2Df.

Now assume that a continuous frequency-domain signal occupies the frequency band f₀ - B < f < f₀ + B, where f₀ >> B. Applying this signal as input to a nonlinear device means that odd-order nonlinearities generally spreads the spectrum at the output. An mth (odd) order nonlinearity results in an output spectrum that occupies the frequency band f₀ - m B < f < f₀ + m B no matter how small B is"this is due to spectrum folding. Next we have the Fourier series representation used for stochastic signal description. Assume for now that B is extremely small, for example, B = 0.5 Hz. Intuitively, it should be sufficient to represent this very narrow-band signal by a single frequency component at f₀, which represents the signal in the frequency band f₀ - B < f < f₀ + B. Since this signal is represented by a single frequency component, the output from an mth (odd) order nonlinearity also occupies the frequency band f₀ - B < f < f₀ + B. Compared to the continuous frequency-domain signal, this is obviously incorrect. The conclusion of this analysis is that no matter how small the input bandwidth, a "number" of discrete frequency points must always be used to obtain a fair representation of the output signal from a nonlinear device. Figure 1b illustrates the bandwidth problem for a stochastic and discrete spectrum approach with three frequency points representing the input signal. The lower and upper spectral regrowth sidebands each have a bandwidth of 2Df, whereas the correct bandwidths are 3Df.

Figure 1: Output spectrum of a third-order non-linearity. (a) An input consisting of three cosine tones with frequencies (f₁, f₂, f₃) along with their inter-modulation components. (b) The power spectral density representing an arbitrary signal, where each frequency component represents the signal in a Df bandwidth.

Assume the input signal is represented at #_Q frequency points such that 2B = x #_Q, where x is the frequency resolution in the discrete spectrum approximation. This means that the output signal from an mth (odd) order nonlinearity occupies the frequency band . Thus, the discrete spectrum bandwidth is 100 (m-1)/(m#_Q) percent too low. As shown in Figure 2, the percentage error in bandwidth is quite large"even for high Q's. Also notice from Figure 2 that the order dependence only has minor impact on bandwidth error.

Figure 2: Percentage bandwidth error between the approximate discrete spectrum approach and the true continuous spectrum approach.

Using a discrete-bandwidth analysis introduces a frequency bandwidth error. There is also some error in the power estimation, since a discrete spectrum is approximating a continuous spectrum"Figure 3 illustrates this concept. The figure is calculated using the techniques described in . In Figure 3, we use only three frequency points to describe a CDMA signal with a bandwidth of 1.25 MHz. The technique by Wu et al. is the reference for this operation, since it includes the correct spectral folding of the continuous spectrum signals. As shown in Figure 3, the discrete spectrum approach does not predict the correct bandwidth, as discussed in the previous section of this article.

Figure 3: Power spectral density for the output of a nonlinear power amplifier for IS-95 applications.

However, there is also an error in amplitude"in particular at the lower and upper edges of the spectral regrowth. For the discrete spectrum analysis in , the predicted total 3rd-order spectral regrowth power is shown in Figure 4, along with the reference set by the Wu technique. The percentage error in total power of the lower/upper regrowth sidebands is shown in Figure 5. Figures 4 and 5 show that the power predicted by the discrete spectrum approach rapidly converges towards the reference for an increasing number of frequency points in the discrete-spectrum representation. Figure 5 shows that the error is less than 1% for Q's above 5.

Figure 4: The total spectral regrowth powers in both lower and upper adjacent sidebands.

Figure 5: Percentage error in total upper/lower sideband power between the discrete spectrum approach and the continuous spectrum approach.

This article has shown that care must be taken when using discrete-spectrum techniques to analyze spectral regrowth in nonlinear devices. Using too few frequency points to represent an otherwise continuous spectrum signal gives an underestimation in output bandwidth and, more importantly, an error in power level. We show an example to illustrate these concepts for spectral regrowth in an IS-95 CDMA power amplifier. The example shows that using 11 frequency points in the discrete spectrum approach, gives a bandwidth error of approximately 6% whereas the spectral regrowth power error is well below 1%. Since power estimation is the most important parameter, this is considered to be acceptable. The example can be considered typical in the sense that third-order nonlinear effects are dominated by the first-order (linear) effect. When choosing the number of frequency points, you need to consider two issues:

1. The bandwidth error is usually unproblematic in practical situations, but it is important to be aware of the phenomenon.
2. Choose the number of frequency points in the discrete-spectrum technique such that the spectral regrowth power has converged. You can easily achieve this convergence by analyzing cases with different numbers of frequency points.